When required to identify the orientation of an item outside the center of the visual field, the mean orientation predicts performance better than the orientation of any individual item in that region. Here I examine whether the visual system also preserves the variance of orientations in these so-called “crowded” displays. In Experiment 1, I determined the separation between items necessary to prevent neighbors from interfering with discrimination between different orientations in a single, target item. In Experiment 2, I used this separation and measured the effect of orientation variance on discrimination between mean orientations in these consequently uncrowded displays. In Experiment 3, I measured the relationship between the just-noticeable difference in variance and the smaller of two orientation variances in uncrowded displays. Finally, in Experiments 4 and 5, I reduced the separation between items and measured the effect of crowding on mean and variance discriminations. When considered together, the results of all these experiments imply that the visual system computes orientation variances with both more efficiency and greater precision than it computes orientation means. Although crowding made it difficult for some observers to discriminate between small amounts of orientation variance, it had no other significant effect on visual estimates mean orientation and orientation variance.

^{1}

*σ*) of 0.25 degree of visual angle. Both grating and blob had maximum contrast.

*σ*= 0.125°). The pre-cue had the same azimuth as the target, but its retinal eccentricity was greater: 6.5 degrees of visual angle. The 0.1-s cue–target onset asynchrony seems to be optimal for automatically attracting attention (Cheal & Lyon, 1991; Nakayama & Mackeben, 1989).

*θ*

_{1}− 11.2°,

*θ*

_{1}− 5.6,

*θ*

_{1}+ 5.6°,

*θ*

_{1}+ 11.2°}, where

*θ*

_{1}denotes the orientation of the first target. The observer's task was to report whether the second target was oriented clockwise or anti-clockwise of the first target. Pilot experimentation suggested that accuracy with these orientations would fall within the interval (50%, 99%). Yet another pilot experiment established accuracies in excess of 98% correct when arrays contained just one Gabor, and orientations were selected randomly from the set {

*θ*

_{1}− 45°,

*θ*

_{1}+ 45°}.

_{1}, was selected randomly from the interval [0°, 180°). The Gaussian distribution of Gabors in the second array had the same standard deviation, but its mean

_{1}± Δ

*σ*

_{G}∈ {1°, 2°, 4°, 8°, 16°}) were paired with four numbers of Gabor per array (

*N*∈ {1, 2, 4, 8}). The staircases were optimized to converge on the mean

*α*of a (Weibull) psychometric function having the form

*P*(C) represents the probability of a correct response. The value used for psychometric slope

*β*= 10 was the arithmetic mean of maximum likelihood fits of Equation 1 to JAS's data from each condition in Experiment 1.

*N*. These symbols have been nudged horizontally for maximum legibility. All observers suffered an elevation of JND when the standard deviation of Gabor orientations was highest, but no systematic effect of array size is obvious from a visual inspection of the data.

*σ*

_{G}is the standard deviation of the Gabor orientations, Φ is the cumulative Normal distribution function, and

*M*

_{mean},

*σ*

_{E}, and

*σ*

_{L}are free parameters.

*M*

_{mean}represents the number of Gabors observers use when calculating the mean of each array.

*M*

_{mean}is necessarily an integer and

*M*

_{mean}≤

*N*.

*σ*

_{E}and

*σ*

_{L}represent the standard deviations of the early and late noises, respectively. Psychometric floor

*γ*and ceiling 1 −

*δ*were set to reflect the empirical lapse rates:

*γ*=

*δ*= 0.03 for HLW,

*γ*=

*δ*= 0.04 for MJM and AS, and

*γ*=

*δ*= 0.05 for JAS. NB: The fraction in Equation 2 has a 2 in its denominator because of the 2AFC paradigm; decisions are necessarily affected by orientation variance in both arrays.

*M*

_{mean}was allowed to vary with array size

*N,*all data from observer MJM were nonetheless best fit when

*M*

_{mean}= 1. In this case (i.e., when

*M*

_{mean}does not vary with

*N*), one of the two noise parameters is redundant. Therefore, I arbitrarily set

*σ*

_{L}= 0 and found that likelihood was maximized when

*σ*

_{E}= 4.9°. Fits to JAS's data were best when

*σ*

_{E}= 4.0°,

*σ*

_{L}= 8.8°, and

*M*

_{mean}=

*σ*

_{E}= 2.9°,

*σ*

_{L}= 4.5°, and

*M*

_{mean}=

*σ*

_{E}= 2.1°,

*σ*

_{L}= 5.2°, and

*M*

_{mean}=

^{2}

*M*

_{max}uncrowded Gabors, when attempting to compute their mean orientation. In other words, the fits satisfy the constraint

*M*

_{mean}= min{

*M*

_{max},

*N*}, where

*M*

_{max}is an arbitrary integer, representing the maximum sample size. Imposition of this constraint results in an insignificantly poorer fit to HLW's data, as confirmed by the generalized likelihood ratio test (Mood, Graybill, & Boes, 1974).

^{3}

*M*

_{max}≥ 8). Its fit to MJM's data is clearly inferior, but its fit to the other observers' data at least does not appear entirely awful. To determine which capacities can be safely ruled out by each observer's data, I once again turned to the generalized likelihood ratio test. The results indicate that we can reject the hypothesis that

*M*

_{max}> 1 for MJM, we can reject the hypothesis that

*M*

_{max}> 4 for JAS, and we can reject the hypotheses that

*M*

_{max}> 3 for both AS and HLW.

*M*

_{mean}≈

*M*

_{mean}much closer to the array size

*N*. As noted above, Experiment 2 differs from previous work by using a uniform distribution of mean orientations. If observers had different sampling efficiencies for different mean orientations, perhaps the small values of

*M*

_{mean}suggested by Experiment 2 were simply an artifact of collapsing across different mean orientations.

*M*

_{mean}.

*variance*discrimination not mean discrimination.

*N*= 8 were used in this experiment. As before, the orientations in each array were selected from a Gaussian distribution. However, in this experiment, the mean of the Gaussian distribution was reselected at random for every array (not merely every trial). Within each trial, the distribution defining one array had a pedestal variance that was randomly selected from the set {1 deg

^{2}, 4 deg

^{2}, 16 deg

^{2}}. The variance of the distribution defining the other array was determined by QUEST. It was the observer's task to select this latter array.

*P*(C) = 0.67, the other converging on an accuracy of

*P*(C) = 0.84, assuming Weibull psychometric functions like that defined in Equation 1 and substituting the increment in variance Δvar

*θ*(in deg

^{2}), for the increment in mean Δ

*β*= 3 was determined in a pilot experiment.

*θ*was set to a value of 900 deg

^{2}, was 0.10. This seemed sufficient for a reasonable estimate of lapse rate.

^{4}Data from IM clearly do not suggest any dip, but data from JAS do (see Figure 5). At least, they contain the hint of a dip: the JND for a 4-deg

^{2}pedestal is slightly lower than that for 1-deg

^{2}pedestal. Confidence intervals suggest that this dip may not be significant, but I applied the sensory threshold model described by Morgan et al. (2008) just to make sure.

*δ*≠ 0. Proportion correct is given by the formula

*F*is the

*F*-distribution, with degrees of freedom

*M*

_{var}− 1 and

*M*

_{var}− 1. As in Equation 2, here

*M*

_{var}represents the number of Gabors observers use when calculating the variance of each array.

*M*

_{var}is necessarily an integer and

*M*

_{var}≤

*N*.

*σ*

_{E}represents the standard deviation of the early noise, and

*δ*is the empirically determined lapse rate (0 for all observers in Experiment 3, except AS, for whom it was 0.014).

*F*-distribution. Obtaining maximum likelihood fits with such a complicated psychometric function would not only be computationally intractable, it would necessarily over-fit the data from Experiment 3, which used just one array size, and thus cannot simultaneously constrain the variances of early and late noises.

*M*

_{var}= 7 and

*σ*

_{E}= 2.5°. For JAS, the best-fitting parameter values were

*M*

_{var}= 6 and

*σ*

_{E}= 2.0°. For AS, they were

*M*

_{var}= 6 and

*σ*

_{E}= 3.5°, and for HLW, they were

*M*

_{var}= 8 and

*σ*

_{E}= 3.0°. A comparison of these values with those derived in Experiment 2 suggests that discriminations of orientation variance enjoy both greater sampling efficiency and less noise than discriminations of mean orientation, at least with uncrowded arrays.

*σ*

_{E},

*σ*

_{L},

*M*

_{mean}, and

*M*

_{var}. After finding the maximum likelihood fit with all four parameters free to vary (JAS:

*σ*

_{E}= 2.0°,

*σ*

_{L}= 8.5°,

*M*

_{mean}= 2, and

*M*

_{var}= 6; AS:

*σ*

_{E}= 3.5°,

*σ*

_{L}= 3.6°,

*M*

_{mean}= 1, and

*M*

_{var}= 6; HLW:

*σ*

_{E}= 3.0°,

*σ*

_{L}= 3.4°,

*M*

_{mean}= 1, and

*M*

_{var}= 8), I independently tested the hypotheses of no late noise (i.e.,

*σ*

_{L}= 0) and equal sampling efficiencies for the two tasks (i.e.,

*M*

_{mean}=

*M*

_{var}). Data from each observer (except IM, who did not participate in Experiment 2) allowed the rejection of both hypotheses.

*σ*

_{L}/

*σ*

_{E}> 1) must be understood as a lower bound on the ratio for uncrowded mean discriminations that would have been obtained with a non-zero late noise for variance discrimination. To fit mean and variance discriminations as well as they were fit above, the presence of such noise would have to be offset by a reduction in the amount of early noise that was theoretically present for both mean and variance discriminations, and this reduction in turn would have to be offset by yet more late noise for mean discriminations.

*N*= 8 were used. On each trial, the first array was centered at a random azimuth and 6 degrees of visual angle. Observers were encouraged to maintain fixation, so that Gabors in the second array would have the same retinal positions as Gabors in the first array. There were a total of nine viewing conditions: three standard deviations of orientation (

*σ*

_{G}∈ {1°, 8°, 16°}), fully crossed with three Gabor spacings. In the most tightly crowded condition, the distance between the center of each Gabor and the center of its array was 1.5 degrees of visual angle. Nearest neighbors thus had a center-to-center spacing of 1.2 degrees of visual angle. Nearest neighbor spacings for the lesser crowded conditions were 2.3 and 4.7 degrees of visual angle. This latter spacing, which was 0.92 times that used in Experiments 1– 3, was the maximum with which I could be certain to avoid losing parts of Gabors at the top and bottom of the computer screen.

*γ*=

*δ*= 0.01 for each observer in Experiment 4, except AS:

*γ*=

*δ*= 0.03. In Experiment 5, the lapse rates (

*δ*) were IM: 0.02, JAS: 0.02, AS: 0.01, and HLW: 0 (sic).

*M*

_{mean}would be sufficient to describe all mean discriminations and maximized the likelihood of a slightly less general model. In that model, 11 parameters were allowed to vary. They were the sample size

*M*

_{mean}, the late noise that perturbs estimates of mean orientation

*σ*

_{L}, and three parameters for each Gabor spacing: the standard deviation of the early noise

*σ*

_{E}, the sample size used for variance discriminations

*M*

_{var}, and the threshold below which all variances would be indistinguishable

*c*.

*σ*

_{E}would not seriously affect mean discriminations that were primarily limited by late noise. Therefore, I refit each observer's data with a nested model, in which

*σ*

_{E}remained invariant with spacing, and submitted the reduction in likelihood to the chi-square test. The reductions in likelihood were not significant. Therefore, none of the observers' data suggest an effect of Gabor spacing on

*σ*

_{E}.

*c*remained invariant, one in which

*M*

_{var}remained invariant, one in which both remained invariant, and one in which

*M*

_{var}remained invariant with spacing and the sensory threshold

*c*was fixed at zero.

*M*

_{var}to remain invariant with Gabor spacing also resulted in insignificant likelihood reductions when the inefficient, noisy observer model was fit to the data from IM and JAS; however, the other aforementioned nested models were significantly less likely. Therefore, the data from these two observers are consistent with the conclusion that crowding did produce a sensory threshold, but it did not affect the efficiency with which orientation variances were estimated.

*σ*

_{L}= 0 and one for the hypothesis that

*M*

_{var}=

*M*

_{mean}. These constraints also significantly reduced the maximum likelihood. Therefore, all of the data are consistent with the conclusion that orientation-variance discrimination is both less noisy and more efficient than mean orientation discrimination, regardless of the spacing between Gabors.

*σ*

_{E}= 4.0°,

*σ*

_{L}= 3.2°,

*M*

_{mean}= 3,

*M*

_{var}= 8, and sensory thresholds effectively equated all variances below 160 and 240 deg

^{2}, for moderately spaced and tightly crowded Gabors, respectively. For JAS,

*σ*

_{E}= 2.6°,

*σ*

_{L}= 9.9°,

*M*

_{mean}= 2,

*M*

_{var}= 6. Only those data collected with tightly crowded Gabors supported a sensory threshold for this observer, the best-fitting one effectively equated all variances below 89 deg

^{2}. For AS,

*σ*

_{E}= 3.4°,

*σ*

_{L}= 2.2°,

*M*

_{mean}= 2,

*M*

_{var}= 6, and for HLW,

*σ*

_{E}= 2.9°,

*σ*

_{L}= 3.7°,

*M*

_{mean}= 2,

*M*

_{var}= 6. Neither of these latter two data sets supported a sensory threshold.

*M*

_{var}>

*M*

_{mean}.

*range*(i.e., the smallest angle containing all 8 orientations). This range observer's performance is illustrated by the circular symbols in Figure 8b. For comparison, an observer whose mean discriminations are based on the middle of this range is illustrated by the circular symbols in Figure 8a.

*M*

_{mean}.

*σ*

_{G}∈ {1°, 8°, 16°}) × two numbers of Gabor per array (

*N*∈ {1, 8}).

*N*= 1 JNDs and the

*N*= 8 JNDs.

*M*

_{mean}=

*M*

_{mean}= 1∀

*N*; and mean = 67.5°:

*M*

_{mean}=

*M*

_{mean}=

*M*

_{mean}=

*σ*

_{E}= 8.6°).

*M*

_{mean}. The fits also strongly suggest that Experiment 2's randomization of mean orientation in no way contaminated the estimates of sampling efficiency that were inferred from its results.

*σ*

_{E},

*σ*

_{L}, and a separate value of

*M*

_{mean}for

*N*= 2, 4, and 8. (By definition,

*M*

_{mean}= 1 when

*N*= 1.) Since the displays used in this experiment were demonstrably uncrowded, it seems safe to assume that

*σ*

_{E}remains invariant with

*N*. However, it is conceivable that

*σ*

_{L}does not. I have not adopted the assumption that

*σ*

_{L}remains invariant with

*N*in any subsequent modeling, but here it is necessary to avoid “over-fitting” these data with too many free parameters.

*α*was 0.05. Do not confuse this

*α,*preferred by Mood et al. to the somewhat more popular

*P*-value, with the Weibull mean in Equation 1!